Seleucid innovations in the Babylonian "algebraic" tradition and their kin abroad
Høyrup, Jens
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Filosofi og videnskabsteori på Roskilde Universitetscenter, 3. række : Preprints og reprints.
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2000
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Høyrup, J. (2000). Seleucid innovations in the Babylonian "algebraic" tradition and their kin abroad. Filosofi og
videnskabsteori på Roskilde Universitetscenter, 3. række : Preprints og reprints., (1).
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ROSKILDE UNIVERSITETSCENTER
Faggruppen for filosofi og
videnskabsteori
ROSKILDE UNIVERSITY
Section for philosophy
and science studies
SELEUCID INNOVATIONS IN THE
BABYLONIAN "ALGEBRAIC" TRADITION
AND THEIR KIN ABROAD
JENS HØYRUP
FILOSOFI OG VIDENSKABSTEORI PÅ
ROSKILDE UNIVERSITETSCENTER
3. Række: Preprints og reprints
2000 Nr. 1
Published:
Pp. 9–29 in Yvonne Dold-Samplonius et al (eds), From China to Paris: 2000 Years
Transmission of Mathematical Ideas. (Boethius, 46). Stuttgart: Steiner, 2002.
ISSN 0902-901X
The surveyors’ tradition and its impact in
Seleucid innovations . . . . . . . . . . . . . . . .
Demotic evidence . . . . . . . . . . . . . . . . . .
Abū Bakr and Leonardo . . . . . . . . . . . . .
Greco-Roman problems . . . . . . . . . . . . . .
An Indian witness: Mahāvı̄ra . . . . . . . . .
Nine Chapters on Arithmetic . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . .
Mesopotamia
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1
5
12
14
14
16
19
21
The surveyors’ tradition and its impact in Mesopotamia
Much of my work during the last decade or so has dealt with what I have called
the “surveyors’ tradition” and its impact on Babylonian, Greek and Islamic “literate”
mathematics. Most of the results have been published elsewhere,[1] for which reason
I shall restrict this introduction to what is crucial for the theme I have been asked
to deal with.
The tradition in question is a “lay”, that is, non-scribal and for long certainly
also in the main an oral tradition (barring a single known exception, non-scribal
literacy only became a possibility with the advent of alphabetic writing); elsewhere,
when discussing the characteristics of such traditions, I have termed them “subscientific”.[2] Its early traces are found in the Mesopotamian literate record; it is
plausible but not supported by any positive evidence that it was also present in the
Syrian and the core Iranian regions in the second millennium BCE; influence in Egypt
at that date is unlikely.[3] In the later first millennium BCE it influenced both Demotic
Egypt and Greek and Hellenistic mathematics. Whether the altar geometry of the
Śulvasūtras was also influenced is less easily decided (quite apart from the question
whether the constructions described in these first millennium writings go back to
early Vedic, that is, second-millennium practice); however, the indubitable traces
of the tradition in Mahāvı̄ra’s 9th century CE Ganita-sāra-sangraha are likely to go
back to early Jaina times and thus to the late first millennium BCE or the first
centuries CE (see below). Even though the intermediaries are not identified, the
1
A concise presentation will be found in [Høyrup 1997d], a fuller discussion in
[Høyrup 1998] – both concentrating on the proto-algebraic aspects of its material.
Aspects related to practical geometry proper are dealt with in [Høyrup 1997a].
2
See [Høyrup 1990a], [Høyrup 1997c] and [Høyrup 1997b].
3
Firstly, neither the Rhind Mathematical Papyrus nor the Moscow Mathematical
Papyrus betrays any acquaintance with it. Secondly, one problem group in the Rhind
Papyrus (the filling problems, no. 35–38) turns out to be related to (for linguistic
reasons indeed to be derived from) West Asian practitioners’ mathematics (see
[Høyrup 1999a: 124]), indicating that such borrowed problems were not a priori
ostracized, which would otherwise be a possible explanation of their absence from
the two papyri; but this problem type has nothing to do with mensuration. Thirdly,
the problems pointing to West Asia that turn up in Demotic sources are exclusively
related to the innovations of the Seleucid age (see below).
1
Syrian, Iranian and Central Asian regions are unlikely not to have been involved.
Possible influence on the Chinese Nine Chapters on Arithmetic is a question to which
we shall return.
Many traces of the tradition are found in Arabic sources from the later first
and early second millennium CE; some turn up, finally, in Italian abbaco writings
from the late Middle Ages – in part derived from earlier Latin translations of classical
Arabic works, in part however from what must have been still living tradition in
the Islamic world.
One of the Arabic works – perhaps written around 800 CE, perhaps only a witness
of the terminology and thus of the situation of that period, immediately preceding
the work of al-Khwārizmı̄ – is a Liber mensurationum written by one Abū Bakr, so
far known only in a Latin translation due to Gherardo da Cremona[4] (whence the
Latin title). Strangely, it still conserves much of the characteristic phraseology already
found in Old Babylonian (but not late Babylonian) problems derived from the
tradition[5] – so much indeed that transmission within some kind of institutionalized
teaching network seems plausible, perhaps supported by writing since the advent
of non-scribal Aramaic alphabetic literacy. Even though we know nothing about
the geographic origins of Abū Bakr, this consideration seems to locate him (or an
earlier written source which inspired him) in the Iraqo-Syrian area – comparison
with other descendants of the tradition shows that none of these conserves the
original formulations as faithfully as Abū Bakr.
Basic applied mathematics is often so unspecific that shared methods constitute
no argument in favour of transmission or borrowing. Once area measures are based
on the square on the length unit, there is only one acceptable way to find the area
of a rectangle, and only one intuitively obvious way to approximate the area of an
almost-rectangular area (viz the “surveyors’ formula”, average length times average
width). Certain geometric procedures are less constrained by the subject-matter and
may serve, but the best evidence for links is normally supplied by recreational
4
Ed. [Busard 1968]. Abū Bakr’s use of the terms al-jabr and al-muqābalah is pre-al-
Khwārizmı̄an (cf. [Saliba 1972] on the changing use of the two terms); moreover,
murabba is used exclusively in the sense of an unspecified “quadrangle” and speaks
of a square as an “equilateral and equiangular murabba ”, which is another archaic
feature. Abū Bakr may still have written at a later moment, but if so as an exponent
of a tradition whose roots have not been much affected by the innovations brought
about by al-Khwārizmı̄ and Abū Kāmil.
5
See [Høyrup 1986].
2
problems, those mathematical riddles that in pre-modern mathematical practical
professions served to confirm professional identity.
The main evidence for the existence and long-term survival of the surveyors’
tradition is indeed provided by a set of geometrical riddles that turn up in all the
contexts mentioned above, in formulations that often exclude mere transmission
within the literate traditions and from one literate tradition (as known to us, at least)
to another.
The original core of this set of riddles contained a group dealing with a single
square (“???” indicates doubt as to the date from which the problem was present;
s designates the side, 4s “all four sides”, d the diagonal and Q the area of the square;
here and everywhere in the following, Greek letters stand for given numbers):
s+Q = α (= 110)
4
s+Q = α (= 140)
Q–s = α
Q–4s = α (???)
s–Q = α
4
s–Q = α (???)
4
s=Q
d–s = 4 (???)[6]
Other early riddles will have treated of two
squares:
Figure 1. Illustration of the principle
Q1+Q2 = α, s1±s2 = β
Q1–Q2 = α, s1±s2 = β
that the area of a quadratic border
equals
(l,w), where l is its “midlength” and w its width.
When the difference between the areas is given, the two squares were thought of
as concentric and the difference thus as the area of a quadratic border; at least in
later (classical and medieval) times the areas of such quadratic and circular borders
were determined as the product of the “average length” (in the quadratic case 2s1+2s2)
and the width (in the quadratic case (s1–s2)/2) – cf. Figure 1. It would be very strange
if the same intuitively evident rule (a “naive” version of Elements II.8) were not used
in early times.
6
The problem d–s = 4 turns up in the Liber mensurationum in a way which shows
it to be somehow traditional; evidently it refers to the practical assumption that d =
14 if s = 10. But the evidence is insufficient to prove that this was a traditional riddle –
even a traditional approximation might inspire a problem once it was discovered how
to solve it correctly.
3
These problems dealing with a rectangle (length l, width w, diagonal d, area A)
will have circulated in the earliest second millennium:[7]
A = α, l±w = β
A+(l±w) = α, l w = β
A = α, d = β
On circles (with diameter d, perimeter p and area A), at least the problem
d+p+A = α
will have circulated at an early date, probably also the simpler version where A is
omitted.
Most of the problems will have been solved in the original environment in the
same way as in the Old Babylonian school, viz by means of a “naive” (that is, noncritical though reasoned) analytical cut-and-paste technique. When a length and a
width were involved, the solution made use of what we shall call average (µ =
and deviation (δ =
l w
)
2
l–w
) – a technique that is familiar from Elements II.5–6.
2
This restricted set was adopted into the Old Babylonian school not long before
1800 BCE, and developed into a genuine mathematical discipline of algebraic character,
in which measurable segments served to represent entities of other kinds (at times
areas or volumes), and where coefficients were varied freely. But this discipline died
with the school institution itself at the collapse of the Old Babylonian social structure
around 1600 BCE, and only one trace of its sophistication can be found in later sources
(a problem type where the sides of a rectangle represent igûm and igibûm, “the
reciprocal and its reciprocal”, a pair of numbers belonging together in the table of
7
In fact, the two problems A = α, l±w = β are likely to constitute the very beginning
together with the problems A = α, l = β and A = α, w = β. These four constitute a
cluster and still go together in ibn Thabāt’s Reckoner’s Wealth from c. 1200 CE) [ed.,
trans. Rebstock 1993: 124]. The problems A = α, l = β and A = α, w = β are already
found in Akkadian tablets from the 22nd century
BCE,
but the other two not yet
(which is part of the evidence that the trick of the quadratic completion was only
discovered somewhere between 2100
BCE
and 1900
BCE).
One may add that area metrologies were sufficiently complex to make the
apparently innocuous “division problems” A = α, l = β and A = α, w = β unpleasantly
complex.
4
reciprocals). When “algebraic” problems turn up sparingly in Late Babylonian but
apparently pre-Seleucid texts (maybe c. 500 BCE?), they appear to have been a fresh
borrowing from the surveyors’ tradition,[8] and do not go beyond the restricted
range of original riddles.[9] Methods are still traditional, based on naive cut-andpaste geometry and on average and deviation.
Seleucid innovations
Innovation, instead, is unmistakeable in the two Seleucid tablets AO 6484 and
BM 34568.[10] The former of these is from the early 2nd century BCE[11], the latter –
and most interesting – is undated but roughly contemporary.
One problem from the latter text treats of alligation, all the others deal with
rectangular sides, diagonals and areas;[12] apart from determinations of d or A from
l and w or of w from d and l, everything is new in some way. Two problems are
traditional as such, giving A and either l+w or l–w; but the procedures differ from
traditional ways, finding for instance in the former case l–w as (l w)2 – 4A , next
w as
(l w) – (l – w)
, and finally l as (l+w)–w; the method of average and deviation
2
is nowhere used (not even in the reverse version where l and w are found from l+w
and l–w as average and deviation).
The remaining problems belong to totally new types. In total, the content of the
tablet is as follows:[13]
8
In any case, the transmission has not been carried by the Babylonian scribal tradition
proper, since the technical use of Sumerograms in the terminology is discontinuous.
But we cannot exclude that part of the transmission had been carried by peripheral
scribal groups (Hittite or Syrian), as was the case for astrology, cf. [Farber 1993: 253f].
9
See [Friberg 1997]. For supplementary (linguistic) evidence that the texts in question
antedate the Seleucid era, see [Høyrup 1999a: 161].
10
Ed. Neugebauer in [MKT I, 96–99] and Waschow in [MKT III, 14–17], respectively.
11
See [Høyrup 1990b: 347 n.180].
12
One is dressed as a problem about a reed leaned against a wall. In the general
context of rectangular problems it is obvious, however, that the underlying problem
is d–l = 3, w = 9 (symbols as above).
13
I interpret the sexagesimal place value numbers in the lowest possible integer order
of magnitude and transcribe correspondingly into Arabic numerals.
5
(1)
l = 4, w = 3; d is found as ½l+w – first formulated as a general rule, next
done on the actual example.
d2–l2 .
(2)
l = 4, d = 5; w is found as
(3)
d+l = 9, w = 3; l is found as
(4)
d+w = 8, l = 4; solution corresponding to (3).
(5)
l = 60, w = 32; d is found as
(6)
l = 60, w = 32; A is found as l w.
(7)
l = 60, w = 25; d is found as
(8)
l = 60, w = 25; A is found as l w.
(9)
l+w = 14, A = 48; 〈l–w〉 is found as
½ ([d l ]2 – w 2)
, d as (d+l)–l.
d l
l 2 w 2 = 68.
l 2 w 2 = 65.
(l w)2 – 4A = 2, w as
½ ([l+w]–〈l–w〉) and l finally as w+〈l–w〉.
(10)
l+w = 23, d = 17; 〈2A〉 is found as ([l+w]2–d2) = 240, 〈l–w〉 next as
(l w)2 – 4A = 7 – whence l and w follow as in (9).
(11)
d+l = 50, w = 20; solved as (3),[14] l = 21, d = 29.
(12)
d–l = 3, w = 9 (the reed problem translated into a rectangle problem, cf.
½ (w 2 [d – l]2)
note 12); d is found as
= 15, l as
d–l
(13)
d+l = 9, d+w = 8; 〈l+w+d〉 is found as
d 2 – w 2 = 12.
(d l)2 (d w)2 – 1 = 12, where 1
obviously stands for (l–w)2 = ([d+l]–[d+w])2; next, w is found as
〈l+w+d〉–(d+l) = 3, d as (d+w)–w, and l as (d+l)–d.
Entities that are found but not named in the text are marked 〈 〉.
14
With the only difference that the division in (3) is formulated as the question “9
steps of what shall I go in order to have 36”, whereas the present problem multiplies
by the reciprocal of d+l. Since this method is distinctively Babylonian and thus
irrelevant for the questions of borrowing and influence I shall not mention it further
on.
6
(14)
l+w+d = 70, A = 420; d is found as
½ ([l w d ]2 – 2A)
= 29.
l w d
(15)
l–w = 7, A = 120; 〈l+w〉 is found as
(l – w)2 4A = 23, w as
½ (〈l+w〉–[l–w]) = 8, l as w+(l–w).
(16)
A cup weighing 1 mina is composed of gold and copper in ratio 1:9.
(17)
l+w+d = 12, A = 12; solved as (14), d = 5.
(18)
l+w+d = 60, A = 300; not followed by a
solution but by a rule formulated in
general terms and corresponding to
(14) and (17).
(19)
l+d = 45, w+d = 40; again, a general
rule is given which follows (13).
(Several problems follow, too damaged
however to allow interpretation)
Though not occurring as problems of the simple
form of (2), (5) and (7), the Pythagorean theorem
(or, as I shall prefer to call it in a context where
“theorems” have no place, the “Pythagorean
Figure 2. The diagram underlying BM
rule”) is familiar from the Old Babylonian cor- 34568, no. (9) and (15).
pus.[15] So is of course the area computation of
(6) and (8). No. (2), (5), (6), (7) and (8) thus present us with no innovation beyond
the numerical parameters.
The use of a special rule for the case where l and w are in ratio 4:3, on the other
hand, is certainly an innovation, even though this ratio was the standard assumption
of the Old Babylonian calculators and the reason that 1;15 (= 1¼) appears in tables
of technical coefficients as “the coefficient for the diagonal of the length and
width”.[16] In later times, a standard rule connected the ratios 3:4:5 with constant
15
[Høyrup 1999b] presents a complete survey of its published appearances.
16
TMS III, 35, ed. [Bruins & Rutten 1961: 26]. “Length and width” stands for the
simplest configuration determined by a single length and a single width, that is,
the rectangle.
7
differences;[17] it is a reasonable assumption that
(1) is a first extant witness of this rule.
As problems, (9) and (15) are familiar from Old
Babylonian texts, and they are likely to represent
the very beginning of the tradition for mixed
second-degree riddles – cf. note 7. The solutions,
however, are not the traditional ones based on
average µ =
l w
l–w
and deviation δ =
; instead,
2
2
they follow the diagram of Figure 2, which is also
likely to have been the actual basis for the reasoning. It is a striking
stylistic
feature
Figure 3. The reed leaned against
(and also a devithe wall.
ation from earlier
habits) that even the possibility to determine l and
w by a symmetric procedure (namely as
and
(l w) (l – w)
2
(l w) – (l – w)
, respectively) is not used.
2
Problems treating of a reed or pole leaned against
the wall are already present in an Old Babylonian
anthology text BM 85196[18]. The situation is shown
Figure 4. A geometric justification
in Figure 3: A reed of length d first stands vertically for the solution of BM 34568 no.
against a wall; afterwards, it is moved to a slanted 12.
position, in which the top descends to height l (the
descent thus being d–l) while the foot moves a distance w away from the wall.
In the Old Babylonian versions, d is given together with either w or d–l. Solution
thus requires nothing beyond simple application of the Pythagorean rule. We cannot
know, of course, whether the problem type of the present tablet (d–l and w given)
was also dealt with in Old Babylonian times; if so, however, the solution would
17
Referred to in the Liber mensurationum [ed. Busard 1968: 97] and in Leonardo’s
Pratica [ed. Boncompagni 1862: 70].
18
Ed. Neugebauer in [MKT II, 44]. On the wide diffusion of this problem type, see
[Sesiano 1987].
8
probably have taken advantage of the facts that
(w) =
(d)– (l) =
(2l+2w,
l–w
) – possibly
2
expressed as (l+w,l–w), cf. above, p. 3 – and thus
have found d+l as w2/(d–l).[19]
This, as we see, is not done here. In algebraic
symbolism, the solution follows from the observation that (d–l)2+w2 = d2+l2–2dl+w2 = 2d2–2ld =
2d (d–l). It can also be explained from the diagram
of Figure 4: (d–l)2 corresponds to the area Q, whereFigure 5. The geometric justification
as w2 corresponds to (d)– (l) = (d)–S = 2P+Q. The
for the case l+w+d = α, A = β.
sum of the two hence equals 2P+2Q = 2 (d,d–l).
In itself, this is nothing but a possible basis for the argument, though supported by
the fact that halving precedes division by d–l, which
makes best sense if the doubled rectangle is reduced
first to a single rectangle; seen in the light of what
follows imminently (and since halving invariably
precedes division by the measure of a side in parallel
cases), this or something very similar seems to have
been the actual argument.
Most in favour is obviously the type represented
by (14), (17) and (18). The procedure can be explained
from Figure 5: (l+w+d)2 is represented by
P+R+T+2A+2Q+2S; removing 2A and making use of
the fact that P+R = T we are left with 2Q+2S+2T = Figure 6. The probable geometrical argument for the type d+l =
2 (d,l+w+d).
α, w = β.
This proof is given by Leonardo Fibonacci in the
Pratica geometrie, in a way which indicates that he has not invented it himself – at
most he has inserted a diagonal (omitted here) in order to explain the construction
19
Here and in the following,
with side s, and
(s) designates the (measured or measurable) square
(l,w) the rectangle contained by l and w.
9
of the diagram in Euclidean manner.[20] In view of the faithfulness of both Leonardo
and Abū Bakr when they repeat the “new” procedures of the Seleucid style (see
below), it is a reasonable assumption that the geometric proofs go back to the same
source.
Of the remaining problem types, two are quite new and one only new as far
as the procedure is concerned. The type of (3), (4) and (11) can be regarded as a
counterpart of (12) (the reed problem). The similarity to this type and to that of (14),
(17) and (18) makes it reasonable to look for an analogous justification – see Figure 6:
the whole square represents (l+d)2; if we notice that T = P+w2 and remove w2, we
are left with 2P+2Q = 2 (l,l+d).
The type of (13) and (19) is likely to be based on a slight variation of Figure 5,
shown here in Figure 7 (which might equally well have served for the type l+w+d =
α, A = β, but which happens not to be the proof given by Leonardo). As we see,
(d+l)2+(d+w)2 = (P+2Q+R)+(R+2S+T) = (l+w+d)2+(R–2A). That R–2A equals (l–w)
was familiar knowledge since Old Babylonian times; it follows from Figure 2 if we
draw the diagonals of the four rectangles A, as shown in Figure 8 (cf. presently);
if we express R as (l)+ (w) it can also be seen from the “naive” version of Elements
II.7 used in Figure 4, which is likely to have been in still longer use.
The last type is (10). In this exact form it is not found in earlier Babylonian texts,
20
Ed. [Boncompagni 1862: 68]. Firstly, it is evident from scattered remarks in the
work that Leonardo renders what he has found; secondly, all steps of the procedure
correspond to what is given in the Liber mensurationum [ed. Busard 1968: 97], but
in the passage in question there is no trace of verbal agreement with Gherardo’s
translation (in other places Leonardo follows Gherardo verbatim, correcting only the
grammar). Moreover, Leonardo’s statement runs as follows:
Si maius latus et minus addantur cum dyametro, et sint sicut medietas aree;
et area sit 48
whereas Gherardo has
aggregasti duo latera eius et diametrum ipsius et quod provenit, fuit medietas
48, et area est 48.
Gherardo’s “medietas 48” instead of “24” is obviously meaningless unless it is already
presupposed that this 48 represents the area – in other words, that the sum of all
four sides and both diagonals equals the area. We may therefore conclude that
Leonardo had access to another version of Abū Bakr’s work, in which however the
proof was given, or to some closely related work.
10
but the closely related problem d = α, A = β is
found in the tablet Db2-146.[21] There it is solved
by subtracting 2A from (d), which leaves (l–w),
cf. Figure 8. l and w are then determined from
and
l w
2
l–w
, that is, average and deviation. This
2
problem recurs with the same procedure in Savasorda’s Collection on Mensuration and Partition (the
Liber embadorum)[22]; Abū Bakr and Leonardo[23]
Figure 7. The probable geometric
use the complementary method and add 2A to (d),
justification for the case d+l = α,
d+w = β.
finding thus l+w. All three go on with average and
deviation.
Our Seleucid text starts by finding 2A,
namely as (l+w)– (d), and next calculates
(l–w) as (l+w)–4A.[24] The rest follows the
asymmetric procedure of (9).
Two other cuneiform problem texts of
Figure 8. The diagram underlying Db2-146
and BM 34568, no. 10.
21
Ed. [Baqir 1962].
22
Ed. [Curtze 1902: 48] (Plato of Tivoli’s Latin translation); ed., trans. [Guttmann
& Millàs i Vallicrosa 1931: 44] (Catalan translation from a somewhat different
recension of the Hebrew text).
23
Ed. [Busard 1968: 92] and [Boncompagni 1862: 64], respectively.
24
This seemingly roundabout procedure is the clearest evidence that the configuration
of Figure 8 is indeed used. From a strictly algebraic point of view, it would be
simpler to find (l–w)2 as 2d2–(l+w)2.
11
Seleucid date are known. VAT 7848[25] contains geometric calculations of no interest
in the present context. AO 6484 was already mentioned. It is a mixed anthology,
which is relevant on three accounts:
1. One of its problems (obv. 12, statement only) is a rectangle problem of the type
l+w+d = α, A = β.
2. It is interested in the summation of series “from 1 to 10”. In obv. 1–2, 1+2+...+29
is found, in obv. 3–4 1+4+...+102 is determined. The latter follows the formula
Qn =
n
i
i 2 = (1
1
1
2
n )
3
3
n
i 1
i.
3.
It contains a sequence of igûm-igibûm problems (see above, p. 4), and thus
demonstrates that the tradition of second-degree algebra had not been totally
interrupted within the environment that made use of the sexagesimal place value
system with appurtenant tables of reciprocals. They differ from the Old
Babylonian specimens by dealing with numbers containing up to four significant
sexagesimal places – no doubt an innovation due to the environment of
astronomer-priests where it was produced,[26] and in which multi-place
computation was routine.
So far there is no particular reason to believe that the other innovations of which
the Seleucid texts are evidence were also due to this environment – nor not to believe
it. Although BM 34568 is theoretically more coherent than AO 6484 it is still a
secondary mixture – as shown by the presence of the alligation problem, which is
certainly no invention of the astronomers.
Demotic evidence
At this point, two Demotic papyri turn out to be informative: P. Cairo J.E.89127–
30,89137–43 and P. British Museum 10520,[27] the first from the third century BCE,
the second probably of early(?) Roman date.
The latter begins by stating that “1 is filled up twice to 10”, that is, by asking
for the sums T10 =
10
i 1
i and P10 =
10
i 1
Ti and answering from the correct formulae
25
Ed. Neugebauer & Sachs in [MCT, 141].
26
The colophon of the tablet tells that it was produced by Anu-aba-utēr, who
identifies himself as “priest of [the astrological series] Inūma Anu Enlil”, and who
is known as a possessor and producer of astronomical tablets.
27
Both ed., trans. [Parker 1972].
12
Tn =
n2 n
n 2
n2 n
)(
)
, Pn = (
2
3
2
– not overlapping with the series dealt with in AO 6484 but sufficiently close in style
to be reckoned as members of a single cluster.[28]
The Cairo Papyrus is more substantial for our purpose. Firstly it contains no
less than 7 problems about a pole first standing vertically and then leaned obliquely
against a wall (cf. above, p. 8 and Figure 3). Three are of the easy type where d and
w given, and solved by simple application of the Pythagorean rule; three are of the
equally simple type where d and d–l are given, and solved similarly (both, we
remember, are treated in an Old Babylonian text). Two, finally, are of the more
intricate type found in BM 34568 (w and d–l given) and solved as there.
Further on in the same papyrus, two problems about a rectangle with known
diagonal and area are found. The solution is clearly related to the problems of BM
34568 (cf. also Figure 8): Addition of 2A to (d) and subsequent taking of the square
root gives l+w, whereas subtraction and taking of the root yields l–w. l and w are
then found by the habitual asymmetric procedure.
These and other Demotic mathematical papyri also contain material that descends
from the Pharaonic mathematical tradition as we know it from the Rhind and
Moscow Papyri. What they share with the Seleucid tablets, however, has no known
Egyptian antecedents (neither in actual content nor in style); in some way it
represents an import (there is no reason to doubt the West Asian origin of the reed
28
All our evidence for this cluster is second-century BCE or later. Yet, since no other
traces of Greek influence is found in these texts, the interest in square, triangular
and pyramid numbers and in the sacred 10 of the Pythagoreans is striking.
Even more striking is the fact that the Demotic determination of T10 as
10 2 10
2
corresponds to an observation made by Iamblichos in his commentary to Nicomachos’s Eisagoge, viz that 1+2+...+9+10+9+...+2+1 = 10×10 [Heath 1921: I, 114], whereas
the Seleucid determination of 12+22+...+102 as (1
1
2
10 )
3
3
10
i 1
i turns up in the
pseudo-Nichomachean Theologoumena arithmeticae (X.64, ed. [de Falco 1975: 86], trans.
[Waterfield 1988: 115]), in a quotation from Anatolios.
More precisely, Anatolios gives the sum as “sevenfold”
10
i 1
i , that is, in a form
from which the correct Seleucid formula cannot be derived – another hint that the
Pythagorean knowledge of the formula was derivative.
13
problem, nor the ultimate descent of the rectangle problems from surveyors’
tradition). It is clear, however, that they represent the new stage at least as well as
the Seleucid texts. This does not prove that Demotic Egypt was the place where the
transformation occurred; but the appearance of the characteristic problems as fully
integrated components of Egyptian scribal mathematics in the third century BCE
makes it implausible that the novelty be due to the environment of Seleucid
astronomer-priests. Indeed, West Asian taxators and surveyors will certainly have
followed the Assyrian and Achaemenid conquerors to Egypt; on the other hand,
the methods of Babylonian mathematical astronomy, when eventually reaching Egypt,
did so in a reduced version which shows them to have been carried by astrological
“low” practitioners, not by the scholar-astronomers that created them.
Abū Bakr and Leonardo
The evidence offered by the Liber mensurationum and Leonardo’s Pratica seems
to speak more generally against a localization of the innovations within the surveyors’
core tradition (however this core looked in the late first millennium BCE – but the
chain that transmitted not only problems but also standard phraseology can
reasonable be regarded as a “core”). With slight variations, Abū Bakr has all, and
Leonardo almost all of the problems from BM 34568.[29] However, all problems
that originated in earlier epochs (given l+w and l–w, given A and l±w, given A and
d, given l+w and d) are solved traditionally, by means of average and deviation;
asymmetric procedures occur only in connection with the definitely new problems.
The general shift to asymmetry which we encounter in BM 3456 and the Demotic
papyri was evidently not accepted in the core tradition, although it accepted the
new problems and did not attempt to reformulate these in more symmetric ways.[30]
29
The “reed problem” occurs in regular rectangle version, the problem d+l = α, d+w =
β as d+w = β, l = w+γ (which explains the error in the formulation of BM 34568,
where (l–w)2 appears as an unexplained 1).
30
In another respect, however, even the new problem types as appearing with Abū
Bakr and Leonardo give evidence of normalization, cf. above, note 20: the problem
l+w+d = 24, A = 48 is formulated in a way which shows it beyond reasonable doubt
to have been derived from a problem 2l+2w+2d = A = 48 (subscript “2” meaning
“both”). This predilection for “the perimeter” in rectangle problems is related to what
circulated in Neopythagorean environments and to what we find with Mahāvı̄ra;
in contrast, all Babylonian sources, Old Babylonian as well as Seleucid, are interested
in “both sides” (cf. also BM 34568 no. 17, l+w+d = A = 12)
14
This would agree badly with emergence of the new problems within the core.
Greco-Roman problems
Relevant material from Greek and Latin sources is extremely scarce. Only two
texts that I know of contain evidence of influence from the new “Seleucid” style.
One is the Latin Liber podismi,[31] the title of which betrays it to be a translation
from a Greek original (or at least to be inspired by a Greek model). One of its
problems deals with the rectangle (actually a right triangle, as preferred in all Greek
sources) with given area and diagonal, and does so in the manner of the Demotic
papyrus, with the small difference that l is found first as ½ (〈l+w〉+〈l–w〉), and w
next as l–〈l–w〉. The next problem is overdetermined, giving A, d and l+w; it first
finds 〈l–w〉 from d2 and A, next l as ½ ([l+w]+〈l–w〉), and finally w as (l+w)–l.
Obviously, the solutions of BM 34568 no. 10 and the Demotic diagonal-area problems
are combined, but with the same variant of the asymmetric method as in the previous
problem.
The other is Papyrus graecus genevensis 259[32]. It contains three problems
on right triangles:
1.
w=3,d=5
2.
w+d = 8 , l = 4
3.
l+w = 17 , d = 13
The first tells us nothing. The second (identical with BM 34568 no. 4) makes use of
the fact that l2 = d2–w2 = (d+w) (d–w)[33] and finds 〈d–w〉 as l2/(d+w); w is then found
as ½ ([d+w]+〈d–w〉), and finally d as (d+w)–w. The third – identical with BM 34568
no. 10 apart from the numerical parameters – is solved in an algebraically straighter
way (cf. note 24): 〈(l–w)2〉 is found as 2d2–(d+w)2, w as ½ ([l+w]–〈l–w〉), and l finally
as (l+w)–w.
All in all, these Greek problems might belong in the periphery of a cluster where
31
Ed. [Bubnov 1899: 511f]. The two relevant problems are reproduced in [Sesiano
1998: 298f].
32
Ed. [Sesiano 1999].
33
The trick, we notice, which was not used in the Seleucid reed-problem, cf. p. 9.
On the other hand, it appears to be used in the “pre-Euclidean” version of the
computation of the height of a scalene triangle (see [Høyrup 1997a: 82]), and in
problems about two concentrically located squares; it will thus have been quite
familiar (cf. also the version presented by Elements II.8).
15
the Demotic and Seleucid texts represent something closer to the core: they deviate
slightly in their choice of actual procedures, but the general tenor is the same. They
share the asymmetric approach, but may have replaced the geometric visualization
by more genuine algebraic manipulation. Further, they are formulated in terms of
right triangles instead of rectangles.
An Indian witness: Mahāvı̄ra
Mahāvı̄ra’s 9th century Ganita-sāra-saṅgraha is in itself a late source – slightly
later than al-Khwārizmı̄ and roughly contemporary with Thābit ibn Qurrah.[34]
There are, however, no reasons to doubt Mahāvı̄ra’s assertion that he has taken
advantage of “the help of the accomplished holy sages” when “glean[ing] from the
great ocean of the knowledge of numbers a little of its essence [...] and giv[ing] out
[...] the Sārasaṅgraha, a small work on arithmetic”,[35] that is, that he presents what
was since times immemorial part of the Jaina tradition; moreover, internal evidence
also speaks in favour of the claim.[36] Given the history of the Jaina community,
the adoption of material from the Near East or the Mediterranean region of which
Mahāvı̄ra’s work bears witness is likely to have taken place in late pre-Christian
or at most early Christian centuries.
As a matter of fact, we should probably rather speak of “adoptions” in the plural.
Mahāvı̄ra divides his chapter on geometry into four sections: “approximate measurement”; “minute accurate calculation”; “devilishly difficult problems”; and one on
the “Janya operation”, which does not concern the present argument. All groups
encompass material that is not known from non-Indian sources, but all also contain
rules that are familiar from the Near East and the Mediterranean.
In the section on “approximate” area measurement (pp. 187–197, stanzas VII.7–48)
we find the “surveyors’ formula”; the determination of the area of a circular ring
as average circumference times width (indeed exact); the rule that the circular circumference is three times the diameter; and the problem of finding the separate value
of circumference, diameter and area from their sum (in this order, the characteristic
order of the pre-Old-Babylonian riddle tradition) together with a corresponding rule
(based on the choice of the circumference as the basic parameter and on π = 3).
The section on “minutely accurate” area determination (pp. 197–208, stanzas
34
I am grateful to Yvonne Dold-Samplonius for having first directed my attention
to this important source.
35
Ed., trans. [Raṅgācārya 1912: 3]. All subsequent page references are to this
translation.
36
See the discussion in [Høyrup 1998: 53f], which I shall not repeat here.
16
VII.49–89½) gives the “pre-Euclidean” method for finding the (inner) height in a
scalene triangle, together with Hero’s formula for triangular and quadrangular areas
(for the latter set forth as if it were of general validity); gives the circular area as
one fourth of the product of arc and diameter;[37] and states the circumference to
be √10 times the diameter.
Among the “devilishly difficult” problems (pp. 220–257, stanzas 112½–232½)
we find several of the characteristic traditional surveyors’ riddles (together with
variants with “non-natural” coefficients corresponding to transformation of the
tradition within an institutionalized school environment): area equal to perimeter
(for squares and rectangles);[38] and later on, rectangular area and perimeter given
(a slight variation of the problem A = α, l+w = β); rectangular perimeter and diagonal
given (equivalent to problem 10 of BM 34568 and problem 3 of the Geneva Papyrus);
and rectangular area and diagonal given. The perimeter-and-diagonal problem has
the same parameters as the Geneva version and, more strikingly, finds (l–w)2 in the
same way, viz as 2d2–(l+w)2; the area-and-diagonal problem finds both l+w and l–w,
as do the Cairo Papyrus and the Liber podismi. Even the area-and-perimeter problem
goes via l+w and l–w, not via average and deviation. But in contrast to all the western
versions, l and w are then found by a symmetric procedure with a technical name
of its own (saṅkramana), as average and deviation of l+w and l–w. As in the Seleucid
and Demotic problems, rectangles and not right triangles are concerned.
The riddles have thus been adopted in “Seleucid-Demotic”, not “Old Babylonian”
version; in some respects, however, they echo the Greek rather than the Near Eastern
form. The “minutely accurate” calculations are related to developments that will
only have taken place in the Near East around the mid-first millennium BCE.[39]
37
This formula is already used in Old Babylonian material, but only for the semicircle
where both arc and diameter are external measures.
38
Of these, only the rectangular variant A = l+w is found in the Old Babylonian
material. However, they certainly circulated in the Mediterranean region during the
classical epoch: the Theologoumena arithmeticae mentions repeatedly that the square
(4) is the only square that has its area equal to the perimeter (II.10, IV.23 [ed. de
Falco 1975: 11, 29], trans. [Waterfield 1988: 44, 63]); the second passage cites
Anatolios), doing so in a way that demonstrates this to be a traditional observation;
Plutarch on his part refers to Pythagorean knowledge of the equality of area and
perimeter in the rectangle
(3,6) (De Iside et Osiride 42, ed., trans. [Froidefond 1988:
214f]).
39
The determination of the height of the scalene triangle must predate Euclid, since
17
Whatever was borrowed for this section will thus have arrived when the Jaina school
was already established; but its arrival may well have preceded that of the “devilishly
difficult” problems. Those of the “approximate” rules that were adopted from
elsewhere, on the other hand, may have arrived long before the appearance of
Jainism.
Chapter VI of Mahāvı̄ra’s work contains a section on the summation of series
(pp. 168–176, stanzas 290–317) which at first looks similar to the Seleucid-Demotic
cluster: arithmetical series, geometrical series, and series of the form
i
Ni , where
Ni are squares, cubes or triangular numbers and i runs through an arithmetical series
(with any starting point, any number of members and any difference). What Mahāvı̄ra
offers is, however, so much more elaborate than what we find in the Near Eastern
texts, and its formulas so different, that inspiration one way or the other becomes
a gratuitous hypothesis.
Other Indian sources confirm that such series were a much more serious concern
in India at least after c. 500 CE than they seem to have been in the Seleucid-Demotic
area; indeed, both Brahmagupta and Bhaskara II deal with the same types as
Mahāvı̄ra, and both know[40] that
Pn =
n 2
Tn , Qn =
3
1 2n
Tn .
3
Āryabhata I followed by Bhāskara I[41] give slightly different variants, which
however (given the testimony of Brahmagupta and Bhaskara II) are likely to be
reformulations of the same formulae. All, however, determine Tn from the general
Elements II.13 can reformulate it and II.12 extend it to the case of the outer height;
but the absence from the pre-Seleucid but still Late Babylonian tablets published
in [Friberg, Hunger & al-Rawi 1990] and [Friberg 1997] shows that the invention
cannot predate 500
40
BCE
by much, if at all.
Trans. [Colebrooke 1817: 290–294] and [Colebrooke 1817: 51–57], respectively. Both
also know that Cn =
n
i 1
i 3 = Tn2, a formula that also turns up in al-Karajı̄’s Fakhrı̄
[Woepcke 1853: 61], in a context which suggests this formula to have belonged to
the same cluster.
41
Āryabhatı̄ya II.21–22, ed., trans. [Clark 1930: 37]; I thank Agathe Keller for giving
me access to her as yet published work on Bhāskara’s commentary [Keller 2000].
18
formula for the sum of an arithmetical series, S
[
(n – 1) d
a] n , n being the number
2
of terms, a the first term, and d the difference – thus as the average term multiplied
by the number of terms, as they indeed explain. This formula is also used in the
Bakhshālı̄ manuscript (likely to be somewhat earlier than Brahmagupta),[42] from
which the more complex sums (Pn, Qn, Cn) are all absent. The formula for S is
obviously based on a purely arithmetical consideration, whereas the DemoticSeleucid-Pythagorean formula appears to be derived from considerations based on
psephoi (cf. note 28). All in all, independent development in the two areas followed
by cross-fertilizations appears to be the most plausible explanation.
Nine Chapters on Arithmetic
The Chinese Nine Chapters on Arithmetic[43] are roughly contemporary with our
Seleucid and Demotic sources. Though in the main a witness of an independent
tradition they do contain problems that appear to point to the Near East and the
Mediterranean – not least a version of the “reed against the wall” in IX.8.
On a first inspection, the whole sequence IX.6–13 looks as if it were related to
the Seleucid and Demotic rectangle problems, even though formulated in varying
dress and mostly so as to deal with right triangles (the right triangle is indeed the
topic of the chapter as a whole). On closer inspection, however, the evidence turns
out to be inconclusive.
Translated into the usual l,w,d symbolism – that is, expressed in the way which
will make kinship stand out as clearly as possible – the problems and their solutions
are the following:
(6)
d–l = 1, w = 5 (the analogue of the Seleucid reed problem BM 34568 no. 12).
l is found as
(7)
w 2 – (d – l )2
½ (w 2 [d – l ]2 )
, whence d, where BM 34568 finds d as
d–l
2 (d – l )
and next l. We observe that the Chinese procedure does not halve before
dividing by (d–l), which suggests that a geometric justification, if once
present, had been forgotten.
d–l = 3, w = 8. The structure of the problem is the same, but the solution
proceeds differently: 〈d+l〉 is found as
w2
d–l
〈d l〉 (d – l)
, and d as
d–l
½ (〈d+l〉+[d–l]). This is related to the second problem of the Geneva papyrus,
42
Ed.,trans. [Hayashi 1995: 439 and passim]
43
Ed., trans. [Vogel 1968].
19
(8)
(9)
(10)
(11)
but different from everything in the Seleucid and Demotic texts (cf. also note
33).
d–l = 1, w = 10. Same problem type and same procedure as no. 7.
Another variation of no. 7.
Yet another variation of no. 7.
d = 100, l–w = 68. 〈
½ (〈
l w
〉 is found as
2
d2–2 (
2
l–w 2
)
2
, and w then as
l w
l–w
〉+〈
〉) – certainly not Seleucid-Demotic in style with its use
2
2
of average and deviation, nor however similar in detail to anything from
the older Near Eastern tradition.
(12)
d–l = 2, d–w = 4. The solution builds on the observation that
(d–[d–l]–[d–w]) = 2(d–l) (d–w). This can be argued from a diagram similar
in style to Figure 7, which served for the problem type d+l = α, d+w = β –
namely the one shown in Figure 9: the full square (d) must equal the sum
of the squares (l) and (w); therefore, the overlap S = (d–[d–l]–[d–w]) must
equal the area which they do not cover, that is, 2R = 2 (d–l,d–w).[44]
(13)
d+l = 10, w = 3. Solved as no. 2 of the Geneva papyrus: [d–l] is found as
w2/(d–l), and l as ½ ([d+l]+[d–l]).
Similarities are certainly present, and the appearance of the pole leaned against the
wall seems to suggest that contact has played a role. But the similarities are always
relative; if the whole interest in this problem type and the approach used in the
solution is ultimately inspired from abroad (which does not follow from a plausible
borrowing of the pole-against-wall dress), then the use of average and deviation
in no. 11 and the predominant use of the identity w2 = (d+l) (d–l) would rather point
44
It should be observed that we are in the case of constant differences, d–l = l–w.
The Chinese solution, of course, does not take advantage of that, and we might
therefore claim that the closest kin in the western sources is the case of arbitrary
differences. This case is not treated by Abū Bakr (nor in the ancient texts); it is treated
by Leonardo [ed. Boncompagni 1862: 71], but he solves it by means of algebra, not
by a rule or a diagram pointing toward the tradition. Even though the Chinese
solution may build on a geometric argument similar to the ones known from
Leonardo (etc.), nothing allows us to connect the actual problem or procedure to
Near Eastern or Mediterranean texts.
20
to a contact preceding the Seleucid epoch and
ensuing independent development – but even that
is certainly no necessary conclusion.
It is highly risky to base the construction of a
stemma on the transformations undergone by a
single phrase, in particular when crosswise contamination is possible. If we are none the less
tempted to engage in a similar perilous game we
may notice that those of our sources that make
preferential use of the identity w2 = (d+l) (d–l) are
Figure 9. The possible geometric
those that formulate problems in terms of right basis for problem IX.12 of the Nine
triangles, not quadrangles; if not allowing any Chapters.
positive conclusions the observation may at least
remind us that the “Seleucid” innovations need not have emerged together, however
much the scarcity of sources tempts us to see them as belonging together.
This observation may be generalized into a conclusion: the written sources that
reflect diffusion of the particular “Seleucid” problem types outside the Near Eastern
area are, like the sources from the Near East itself, too few and too diverse to allow
any certain conclusions concerning the details of the transmission pattern.
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